Hi,
This is a yes and no answer.
In Mountain Lion you have Messages.
It has an iMessage component - in fact when you start it up for the first time you are encouraged to add an Apple ID to create or register and iMessages account.
The same Apple ID can then be used in Messages on the iPhone.
Depending which way round you set this up the original will get a pop up saying the other device(s) is using the Apple ID and the iPhone Number.
Accepting the pop add the relevant "Aliases" to the Receive and Send parts of the individual apps.
When you have created this "Start Point" then iMessages sent to your iPhone or to the Apple ID will be synced to the Mac version.
The same is true of iMessages sent to the Mac (actually people are sending to you as identified by both iPhone Number and Apple ID) and these sync in the other direction to the iPhone.
Messages (SMS) from non iPhone devices will not be synced to the Mac.
Messages prior to the Start Point also will not be synced.
The Mac version has a Save on Closing to store Conversation (iMessages) and Chats (Buddy list as per iChat)
These will reappear like they are chats when you call them up.
They are stored in the .ichat format.
I have not found a way to export the iPhone Messages that have not been deleted.
(You can highlight a whole conversation and then "send" it to your Apple ID but it will show as one long text extract with no indication about who said what and lacks even basics like the bubbles to separate things)
9:00 PM Thursday; May 30, 2013
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.3)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously